Having two wizards is like having two guitarists.
Honestly, I can't think of any class schtick that wouldn't be useful to double. Two armored hard-hitting fighters? That means you can control a wider hallway! Two rogues? Now you can flank from both sides. Double sneak-attack damage in one round! Two clerics means double the healing, double the aiding, double the blessing.
The players just need to agree, out of character, how important it is for them to have unique schticks and be up front about that with each other. Is it fun for two wizards to have a sporting contest to see who can get the fireball off first? Or is one player going to be irked because fireballs were his thing? I have definitely seen situations at my table where players have struggled with this kind of competition (even among two different classes), and when it stops being fun for them, it stops being fun for me. If the new player is an unknown, I can see that being a wild card.
As for inviting new players without consulting the current players. In my experience, I wanted to grow our group (from 4 players, one of whom was super flaky), and I checked with the group before doing so. First off, because I wanted to give them the opportunity to suggest the potential player before putting up a public posting, but also because I felt like they had some stake in the matter. On the other hand, once I started inviting new players, I didn't give my group any say in vetting those players, and we had a couple of months where there seemed to be a rotating slot at our table that kept getting filled by new people. The two new players who ultimately stuck it out are now as much a part of the gaming group as the two who have been involved since day one.